Geodesic
Solutions of x³+y³+z³=nxyz
Acta Arithmetica, Tome 73 (1995) no. 3, pp. 201-213.

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The diophantine equation (1) x³ + y³ + z³ = nxyz has only trivial solutions for three (probably) infinite sets of n-values and some other n-values ([7], Chs. 10, 15, [3], [2]). The main set is characterized by: n²+3n+9 is a prime number, n-3 contains no prime factor ≡ 1 (mod 3) and n ≠ - 1,5. Conversely, equation (1) is known to have non-trivial solutions for infinitely many n-values. These solutions were given either as "1 chains" ([7], Ch. 30, [4], [6]), as recursive "strings" ([9]) or as (a few) parametric solutions ([3], [9]). For a fixed n-value, (1) can be transformed into an elliptic curve with a recursive solution structure derived by the "chord and tangent process". Here we treat (1) as a quaternary equation and give new methods to generate infinite chains of solutions from a given solution {x,y,z,n} by recursion. The result of a systematic search for parametric solutions suggests a recursive structure in the general case. If x, y, z satisfy various divisibility conditions that arise naturally, the equation is completely solved in several cases
DOI : 10.4064/aa-73-3-201-213

Erik Dofs 1

1 
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Erik Dofs. Solutions of x³+y³+z³=nxyz. Acta Arithmetica, Tome 73 (1995) no. 3, pp. 201-213. doi : 10.4064/aa-73-3-201-213. http://geodesic.mathdoc.fr/articles/10.4064/aa-73-3-201-213/

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